Stats without TearsSolutions for Chapter 9

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1
You make probability statements about things that
can change if you repeat the experiment. There’s a 1/6
chance of rolling doubles, because you’ll get doubles about 1/6
of the times that you roll two dice. But
the mean of the population is one definite number. It doesn’t change from one experiment to the next.
Your estimate changes, because it’s based on your
sample and no sample is perfect. But the thing you’re trying to
estimate, mean or proportion, is what it is even though you
don’t know it exactly.

(Statisticians would say, “the population mean or
proportion is not a random variable.” By that, they mean just
what I said in less technical language.)

2

Answer:
A confidence interval for numeric data is an estimate of
the average, and tells you nothing about individuals. Correct
his conclusion to
I’m 90% confident that the average food expense for all TC3 students is between $45.20 and $60.14 per week..

Remark:
Use all or a similar word to show that you’re
estimating the mean for the population, not just the sample of 40
students. There’s no need to estimate the mean of the sample,
because you know the exact sample mean x̅ for your sample.

Remark:
Be clear in your mind that
you’re estimating the average spending per student at
$45–60 a week. Some individual students will quite likely
spend outside that range, so your interpretation shouldn’t say
anything about individual student spending.

3

Answer:
It’s the use of the word average.
When you collect data points that are all yes/no or success/failure,
you have a sample proportion p̂, equal to the number of
successes divided by sample size, and you can estimate a population
proportion. There is no “average” with non-numeric data.

Your 90% confidence estimate is simply
that 27% to 40% usually or always prepare their own food.

Neveready is
95% confident that the average Neveready A cell, operating a wireless mouse, lasts 1711 to 1801 minutes
(28½ to 30 hours).

Common mistake:
Don’t make any statement about 95% of the batteries! Your
CI is about your estimate of one number, the average life of
all batteries.
Your CI has a margin of error of ±15 minutes; the 95% range for
all batteries would be about 4 to 5 hours.

5
(a) p̂ = 5067/10000 = 0.5067

Don’t make the term “point estimate” harder
than it is! The point estimate for the population mean (or proportion,
standard deviation, etc.) is just the sample mean (or proportion,
standard deviation, etc.).

(b) The sample is his actual data, the 10,000 flips.
Therefore the sample size is n = 10,000. The
population is what he wants to know about, all possible flips.
The population size is infinite or “indefinitely
large”.

Common mistake: Don’t say
“n > 30” or “n ≥ 30”.
That’s true, but it doesn’t
help you with binomial data. For computing a confidence interval
about a proportion from binomial data, the “sample size large
enough” condition is at least 10 successes and at least 10
failures, not sample size at least 30.

1-PropZInt 40, 100, .9 →
(.31942, .48058), p̂ = .4

31.9% to 48.1% of all claims at that office have been open for more than a year (90% confidence).

Multiply by s, divide by E, and square the result. This gives
197. But the t distribution is more spread out than the normal (z)
distribution, so you probably want to bump that number up a bit, say
to 200 or so.

You’re
95% confident that the average of all cash deposits is between $179.86 and $198.93.

Common mistake:
Don’t say that 95% of
deposits are between those values — if you look at the
sample you’ll see that’s pretty unlikely. You’re
estimating the average, not the individual deposits in the
population.